Towards The Mathematics of Autonomic Cognitive Organisation

Abstract

Abstract This paper proposes a candidate mathematical basis for the autonomic layer of the Orchard Cognitive Framework. An earlier reduction identified six recurring primitives across seven coupled mechanisms: (P1) typed distinction and non-collapse; (P2) conservative differentiation; (P3) conservative recursive composition; (P4) projection-preserving commutability; (P5) categorical admissibility with conditional alignment cost; and (P6) reciprocal unresolvedness–structure dynamics. Subsequent registers-first reconstruction exposed a deeper question: these verbs repeatedly depend on a carried state comprising provenance, history, standing, constraints, witnesses, claim scope, and temporal validity, while a register substrate without lawful operations is inert. The present paper therefore treats the foundational ordering itself as falsifiable. A temporally situated informational state is provisionally represented as Xₜ = ⟨xₜ, ℛₜ, cₜ, fₜ⟩ with ℛ(Xₜ) = ⟨Πₜ, Hₜ, Σₜ, Λₜ, Wₜ, 𝜍ₜ, Tₜ⟩. Six candidate conservation laws R-C1…R-C6 govern provenance closure, constraint monotonicity, authority transport, witness accountability, temporal honesty, and append-only history. Lawful transport then requires every declared in-scope commitment to be preserved, lawfully transformed under declared loss, explicitly reappraised, or retained as typed unresolvedness; silent disappearance is excluded. A reconstruction programme tests four competing structural hypotheses: registers-first H_R; separable primitives H_P; deeper lawful-transition structure H_L; and decomposition failure H_X. The first adversarial reconstruction result is intentionally incomplete. A situated register-bearing transition space is partially recoverable, but terrain-lawfulness and manifold structure are not. This exposes a located residual, Constitutive Dependence: the possibility that admissibility must depend on carried state rather than merely coexist with it. The candidate law is retained as R-C7 only for successor-basis testing, not retroactively inserted into the frozen basis. The paper further formalises path-dependent trajectory appraisal, a No-Endpoint-Potential Lemma, a non-optimisation axiom, Recursive Liminal Calibration as bounded feedback for improving perceptual honesty without expanding authority, and an empirical programme with killer controls, reconstruction ablations, liveness measures, and time-locked predictions such as SIG-STAIRCASE and SIG-DRAIN. The result is not a completed mathematics of cognition. It is a candidate lawful-transition grammar, a set of conservation commitments, a collection of known residuals, and a research programme designed to make the decomposition lose cleanly if it is wrong. Keywords: AI alignment; cognitive architecture; typed nulls; provenance; lawful transport; abstention; uncertainty; anti-interpolation; recursive composition; constrained optimisation; dynamical systems; representation revision; liveness; Recursive Liminal Calibration. Epistemic status and source discipline The source corpus contains ratified module specifications together with candidate mathematical syntheses, controlled-fixture results, open questions, deferred validations, and newly generated reconstruction artefacts. This paper does not upgrade a claim merely because it appears beside a stronger one. Mark Meaning in this paper [C] Canon-supported lineage: a commitment already present in ratified or accepted Orchard material at its stated scope. [D] Candidate derivation: a mathematical synthesis that appears to follow from source commitments but is not established as a theorem or universal law. [O] Open question: live mathematical or empirical direction whose form remains unearned. [D/O] A usable structural constraint whose literal mathematical interpretation remains open. ✅ / △ / ◇ / ▿ / ⬡ Demonstrated-at-scope / conditional / open / deferred-validation / candidate or parked. Source scope always travels with the claim. Metaphor never changes standing. Terms such as terrain, riverbed, atlas, curvature, stickiness, avulsion, oscillation, and fractalisation may orient an experiment; each mathematical operator must be earned independently. The term “hallucination” is used in the AI-industry sense of plausible but false or unsupported output. Anti-hallucination here is narrower than complete factuality: it denotes resistance to converting unresolved epistemic states into unsupported asserted content.